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Solve the following 2nd order differential equation by finding the complementary function and the particular integral. Simplify the answer by reducing fractional coefficients.

 y′′−2y′−24y=5e3x−9\displaystyle {y}{''}-{2}{y}'-{24}{y}={5}{e}^{{{3}{x}}}-{9} 
  1. Find the complementary function, yc\displaystyle {y}_{{c}}:
    • Characteristic/Auxillary Equation (in terms of m\displaystyle {m}):   (This answer must be an equation.)
    • Solutions to Characteristic Equation:
      m1=\displaystyle {m}_{{1}}=        m2=\displaystyle {m}_{{2}}=  
    • Write the complementary function:
      yc=c1\displaystyle {y}_{{c}}={c}_{{1}}   +c2\displaystyle +{c}_{{2}}  
  2. Find the particular integral, yp\displaystyle {y}_{{p}}:
    • What is an appropriate guess or trial function for the particular integral (in terms of the undetermined coefficients A\displaystyle {A} and B\displaystyle {B})? Do not find A\displaystyle {A} and B\displaystyle {B}.
      yp=\displaystyle {y}_{{p}}=  
    • Find the undetermined coefficients A\displaystyle {A} and B\displaystyle {B}.
      A=\displaystyle {A}=       B=\displaystyle {B}=  
  3. Write the general solution from your work above.
    y(x)=yc+yp=\displaystyle {y}{\left({x}\right)}={y}_{{c}}+{y}_{{p}}=  

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